Remark 4.1 
In section 6 of [1]} we put \(\sharp =\lbrace  h\in \text{all $\alpha _i(h)\in \mathbb {R}$} \rbrace \) ,
so that \( {\scriptstyle \sqrt{-1}}\, \sharp \)  is the standard Cartan subalgebra of the compact
real form \(su_{n+1}\) . Then the integer lattice is
\(I= {\scriptstyle \sqrt{-1}}\, \sharp \cap 2\pi  {\scriptstyle \sqrt{-1}}\, \mathbb {Z}^{n+1}\) . Thus the real roots
\(\alpha ^{\text{real}}=(2\pi  {\scriptstyle \sqrt{-1}}\, )^{-1}\alpha \)  take integer values on the integer lattice, as do all
real weights. The basic real weights are
\(\Lambda _i^{\text{real}}=\tfrac{1}{2\pi  {\scriptstyle \sqrt{-1}}\, }\left(1-\tfrac{i}{n+1}\right)(x_0+\cdots +x_{i-1})-\tfrac{1}{2\pi  {\scriptstyle \sqrt{-1}}\, }\tfrac{i}{n+1}(x_{i}+\cdots +x_{n})\) 
